We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball $B(1)$, as on the Euclidean space $\mathbb{R}^m$. In both cases we obtain several properties of these polynomials, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the Jacobi polynomials on the real line. As in the classical Clifford case, the orthogonality of the polynomials on $\mathbb{R}^m$ must be treated in a completely different way than the orthogonality of their counterparts on $B(1)$. In case of $\mathbb{R}^m$, it must be expressed in terms of a bilinear form instead of an integral. Furthermore, in this paper the theory of Dunkl monogenics is further developed.

A topological space \((X, \tau)\) is a \(D\)-space if for every function \(\varphi\): \(X\rightarrow \tau\) with \(x\in \varphi(x)\) for each \(x\in X\), \(\{\varphi (x): x\in F\}\) covers \(X\) for some closed discrete subset \(F\) of \(X\). The Michael line \(M\), one of the most important elementary examples in general topology, is the Euclidean space \(\mathbb{R}\) isolating the irrationals. In this note we show that (1) the minimal dense linearly ordered extension of \(M\) is hereditarily paracompact, but not monotonically \(D\); (2) the minimal closed linearly ordered extension of \(M\) is monotonically \(D\); (3) if the space \(X\) is a \(D\)-space (resp., a monotone \(D\)-space), then so is its Alexandroff duplicate space \(\mathscr{A}(X)\) and thus \(\mathscr{A}(M)\) is monotonically \(D\).

The notion of orthogonally conformal vector field on a Riemannian manifold is introduced. This class of vector fields properly includes the normalization of nowhere zero conformal ones. It is clarified in several examples. An integral inequality which relates the existence of orthogonally conformal vector fields with properties of the Ricci tensor of a compact Riemannian manifold is proved and some applications are shown.

This paper deals with coefficient estimates for close-to-convex functions with argument $\beta$ ($-\pi/2<\beta<\pi/2$). By using Herglotz representation formula, sharp bounds of coefficients are obtained. In particular, we solve the problem posed by A. W. Goodman and E. B. Saff in [2]$. Finally some complicate computations yield the explicit estimate of the third coefficient.

A new class is introduced consisting of harmonic univalent functions on the exterior unit disk defined by convolution. This class generates several known and new subclasses of harmonic univalent functions as special cases. A necessary and sufficient convolution condition is obtained for functions to belong to the class. A corresponding general class of harmonic functions with negative coefficients is also introduced, and coefficient condition that is both necessary and sufficient is obtained for the class. Extreme points are also determined. As applications, starlikeness conditions of the Liu-Srivastava linear operator involving the generalized hypergeometric functions are discussed.

In the present paper, we obtain an interesting subordination relation for Salagean-type certain analytic functions by using subordination theorem.

Let $E$ be an ideal of $L^0$ over a $\Sigma$-finite measure space $\,(\Omega,\Sigma,\mu)$. For a real Banach space $\,(X,\|\cdot\|_X)\,$ let $\,E(X)\,$ be the subspace of the space $L^0(X)$ of $\mu$-equivalence classes of strongly $\,\Sigma$-measurable functions $\,f:\Omega\rightarrow X\,$ consisting of all those $f\in L^0(X)$ for which the scalar function $\,\|f(\cdot)\|_X\,$ belongs to $\,E$. For a real Banach space $\,Y\,$ a linear operator $\,T:E(X)\rightarrow Y\,$ is said to be order-weakly compact whenever for each $\,u\in E^+$ the set $\,T(\{f\in E(X):\|f(\cdot)\|_X\le u\})\,$ is relatively weakly compact in $\,Y$. In this paper we derive Yosida-Hewitt type decompositions for order-weakly compact operators $\,T:E(X)\rightarrow Y$. In particular, it is shown that if $\,X\,$ is an Asplund space, then an order-weakly compact operator $\,T:E(X)\rightarrow Y\,$ can be uniquely decomposed as $\,T=T_1+T_2$, where $\,T_1, T_2$ are order-weakly compact operators, $\,T_1$ is smooth and $\,T_2$ is weakly singular.

A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that they are flat. Next we prove that a tangent developable surface (resp. cylinder, cone) is a constant angle surface if and only if the generating curve is a helix (resp. a straight line, a circle).

We study hereditary properties of the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We consider mainly the case of amalgamated free products, and we show in particular that the double of amenable groups and the amalgamated free products of two amenable groups over a finite subgroup admit such actions.

In this note we study large linear structures inside the set of Jones functions, which is a {\em highly pathological class} of surjective functions. We show that there exists an infinite dimensional linear space inside this set of functions. Moreover, we show that this linear space is isomorphic to \(\mathbb{R}^\mathbb{R}\), that is, it has the {\em biggest} possible dimension. The result presented in this note is an improvement of several recent results in the topic of {\em lineability}.

In this paper, by using an improved inequality, we improve an existence theorem of periodic solutions for second order Hamiltonian system with a $p$-Laplacian. Moreover, an estimate of solutions is also given. Our results improve those in some known literatures.

Given an analytic self-mapping $s$ of the open unit disk $\mathbb D$ and given a Blaschke product $b$ of degree $k$, we present necessary and sufficient conditions for $s-b$ to have exactly $k$ zeros inside $\mathbb D$. As a corollary, we obtain a Carathéeodory-Julia-Wolff type theorem for meromorphic functions of the form $s/b$.

The paper is concerned with the class $X^{(n)}$ consisting of all functions, which are $n$-fold symmetric, convex in the direction of the real axis and have real coefficients. For this class we determine the Koebe domain, i.e. the set $\bigcap_{f\in X^{(n)}} f(\Delta)$, as well as the covering domain, i.e. the set $\bigcup_{f\in X^{(n)}} f(\Delta)$. The results depend on the parity of $n\in N$. We also obtain the minorant and the majorant for this class. These functions are defined as follows. If there exists an analytic, univalent function $m$ satisfying the following conditions: $m'(0)>0$, for every $f\in X^{(n)}$ there is $m\prec f$, and $\bigwedge_{f\in X^{(n)}}\ [k\prec f \Rightarrow k\prec m]$, then this function is called the minorant of $X^{(n)}$. Similarly, if there exists an analytic, univalent function $M$ such that $M'(0)>0$, for every $f\in X^{(n)}$ there is $f\prec M$, and $\bigwedge_{f\in X^{(n)}}\ [f\prec k \Rightarrow M\prec k]$, then this function is called the majorant of $X^{(n)}$. If these functions exist, then $m(\Delta)$ and $M(\Delta)$ coincide with the Koebe domain and the covering domain for $X^{(n)}$, respectively.

A new formulation for a class of stochastic linear complementarity problems (SLCPs) with finitely many realizations is proposed, which reformulates the SLCPs as a system of smoothing equations without any constraints by an NCP function. Then, we extend an one step smoothing Newton method to this formulation. Moreover, we show that this algorithm converges globally and local quadratically under mild assumptions.

In this article we present core theorems for double sequences whose entries are complex numbers. These results extend work of Miller and Patterson dealing with double sequences of real numbers. The proofs in this paper are much more involved then the proofs in the article just mentioned as the convex sets in the plane are, in general, much more involved then the trivial convex sets in the line. We give an answer to the following question. If $w$ is a bounded double sequence with complex entries and $A$ is a $4$-dimensional matrix summability method, under what conditions on $A$ does there exist $z$, a subsequence (rearrangement), of $w$ such that each complex number $t$, in the core of $w$, is a limit point of $Az$?

Riemannian geometry on the space of (continuous) paths in a manifold $M$ has been studied by Cruzeiro and Malliavin. I will use concepts in path space analysis to define a Levi-Civita connection on free loop space, using the $G^0$ metric. A tangent vector $X$ at a loop $\gamma$ is a vector field along $\gamma$ such that $X(s) \in T_{\gamma(s)}M$. Following closely the calculations done by Fang, the Riemannian curvature $R^{LM}$ is given by $R^{LM}(X,Y)Z (\cdot) = R^M(X(\cdot),Y(\cdot))Z(\cdot)$.